Farhna shaikh webinar_graphcoloring
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This document discusses graph coloring and provides examples of how it can be used to model scheduling problems. It begins by defining graph coloring as assigning colors to vertices such that no adjacent vertices share the same color. It then gives examples of coloring simple graphs and discusses the chromatic number. The document also explains how map coloring can be modeled as a graph coloring problem and provides examples of how graph coloring can be used to model exam scheduling, frequency assignment, and register allocation problems.
Farhna shaikh webinar_graphcoloring
- 1. •Graph Coloring 1 Given a graph, colour all the vertices so that two adjacent vertices get different colours. 4/20/2020 Graph Theory
- 2. 2 A B C G1 4/20/2020 Graph Theory A B C 1 A B C 2 1 3- colorable graph A B C 1 2 3
- 3. Definition of Colorable Let n be a positive number. A simple graph is n - colorable if the vertices can be colored using n colors so that no two adjacent vertices have the same color. 34/20/2020 Graph Theory
- 4. 44/20/2020 Graph Theory A B D C A B D C
- 5. 5 2- colorable graph 4/20/2020 Graph Theory A B D C A B D C A B D C
- 6. Optimal Colouring What graphs have chromatic number one? when there are no edges… What graphs have chromatic number 2? A path? A cycle? A triangle? What graphs have chromatic number larger than 2? Definition: min #colors for G is chromatic number, x(G) 4/20/2020 Graph Theory 6
- 7. Simple Cycles even(C 2) = odd(C 3) = 4/20/2020 Graph Theory 7
- 8. n n(K ) = Complete Graphs 4/20/2020 Graph Theory 8
- 9. even(W 3) =odd(W 4) = 5W Wheels 4/20/2020 Graph Theory 9
- 10. 10 P Q RS T ? - colorable graph Poll Question#1: What is the chromatic number of above graph? A. 2 B. 3 C. 4 D. 5 4/20/2020 Graph Theory
- 11. Graph Coloring Problem Consider a fictional continent. 114/20/2020 Graph Theory
- 12. Map Coloring •Suppose removed all borders but still wanted to see all the countries. •1 color insufficient. 124/20/2020 Graph Theory
- 13. 134/20/2020 Graph Theory So add another color. Try to fill in every country with one of the two colors.
- 17. •PROBLEM: Two adjacent countries forced to have same color. Border unseen. 174/20/2020 Graph Theory
- 18. •So add another color: 184/20/2020 Graph Theory
- 19. •Insufficient. Need 4 colors because of this country. 194/20/2020 Graph Theory
- 20. •With 4 colors, could do it. 204/20/2020 Graph Theory
- 21. From Map Coloring to Graph Coloring The problem of coloring a map, can be reduced to a graph-theoretic problem: 214/20/2020 Graph Theory
- 22. For each region introduce a vertex: 22 From Map Coloring to Graph Coloring 4/20/2020 Graph Theory
- 23. For each pair of regions with a positive-length common border introduce an edge: 23 From Map Coloring to Graph Coloring 4/20/2020 Graph Theory
- 24. Map not 2-colorable, so dual graph not 2-colorable: 24 From Map Coloring to Graph Coloring 4/20/2020 Graph Theory
- 25. Map not 3-colorable, so graph not 3-colorable: 25 From Map Coloring to Graph Coloring 4/20/2020 Graph Theory
- 26. Map is 4-colorable, so Graph as well: 26 From Map Coloring to Graph Coloring 4/20/2020 Graph Theory
- 27. Coloring regions is equivalent to coloring vertices of graph. 27 From Map Coloring to Graph Coloring 4/20/2020 Graph Theory
- 28. 284/20/2020 Graph Theory Poll Question#2: Every bipartite graph is 2-colorable. ? A. Yes B. No
- 29. 4/20/2020 Graph Theory 29 Poll Question#3: Every 2-colorable graph is a bipartite graph.? A. Yes B. No A graph is bipartite graph iff it is 2 – colorable.
- 30. Some facts about Graph Coloring 4/20/2020 Graph Theory 30 • x(G) <= |V| • If some subgraph requires some K colors then x(G) >= |K| • x(G) = maximum (x(C) | C is connected component in G)
- 31. Graph Coloring Problem Suppose, want to schedule some final exams for CS courses with following course codes: 1007, 3137, 3157, 3203, 3261, 4115, 4118, 4156 Suppose also that there are no common students in the following pairs of courses because of prerequisites: 1007-3157, 3137-3157 1007-3203 1007-3261, 3137-3261, 3203-3261 1007-4115, 3137-4115, 3203-4115, 3261-4115 1007-4118, 3137-4118 1007-4156, 3137-4156, 3157-4156 How many exam slots are necessary to schedule exams? 314/20/2020 Graph Theory
- 32. •Turn this exam scheduling problem into a graph coloring problem. •Here, vertices are courses, V1->1007, V2->3137, V3->3157, V4->3203, V5- >3261, V6->4115, V7->4118, V8->4156 •And edges are the courses which cannot be scheduled simultaneously because of possible students in common i.e. 1007-3157, 3137-3157 1007-3203 1007-3261, 3137-3261, 3203-3261 1007-4115, 3137-4115, 3203-4115, 3261-4115 1007-4118, 3137-4118 1007-4156, 3137-4156, 3157-4156 4/20/2020 Graph Theory 32
- 34. One way to do this is to put edges down where students mutually excluded… 4/20/2020 34 1007 3137 3157 3203 4115 3261 4156 4118 Graph Theory V1 V2 V3 V4 V6 V5 V8 V7
- 35. …and then compute the complementary graph: 4/20/2020 35 1007 3137 3157 3203 4115 3261 4156 4118 Graph Theory V1 V2 V3 V4 V6 V5 V8 V7
- 36. Redraw: 4/20/2020 36 1007 3137 3157 3203 4115 3261 4156 4118 Graph Theory V1 V2 V3 V4 V6 V5 V8 V7
- 37. Not 1-colorable because of this edge 4/20/2020 37 1007 3137 3157 3203 4115 3261 4156 4118 Graph Theory V1 V2 V3 V4 V6 V5 V8 V7
- 38. Not 2-colorable because of this triangle 4/20/2020 38 1007 3137 3157 3203 4115 3261 4156 4118 Graph Theory V1 V2 V3 V4 V6 V5 V8 V7
- 39. Is 3-colorable. Try to color by Red, Green, Gray. 4/20/2020 39 1007 3137 3157 3203 4115 3261 4156 4118 Graph Theory V1 V2 V3 V4 V6 V5 V8 V7
- 40. V4->3203-Red, V3->3157-Gray, V7->4118-Green: 4/20/2020 40 1007 3137 3157 3203 4115 3261 4156 4118 Graph Theory V1 V2 V3 V4 V6 V5 V8 V7
- 41. So V8->4156 and V5->3261 must be Gray and Red resp. 4/20/2020 41 1007 3137 3157 3203 4115 3261 4156 4118 Graph Theory V1 V2 V3 V4 V6 V5 V8 V7
- 42. So V6->4115 must be Red. 4/20/2020 42 1007 3137 3157 3203 4115 3261 4156 4118 Graph Theory V1 V2 V3 V4 V6 V5 V8 V7
- 43. V2->3137 and V1->1007 are easy to color. 4/20/2020 43 1007 3137 3157 3203 4115 3261 4156 4118 Graph Theory V1 V2 V3 V4 V6 V5 V8 V7
- 44. So need 3 exam slots: 4/20/2020 44 1007 3137 3157 3203 4115 3261 4156 4118 Slot 1 Slot 2 Slot 3 Graph Theory V1 V2 V3 V4 V6 V5 V8 V7
- 45. Homework#1: Flight Gates flights need gates, but times overlap. how many gates needed? 122 145 67 257 306 99 Flights time 4/20/2020 Graph Theory 45
- 46. Conflict Graph 99 145 306 Needs gate at same time • Each vertex represents a flight • Each edge represents a conflict 4/20/2020 Graph Theory 46
- 47. 257 67 9 145 306 122 Graph Colouring There is a k-colouring in this graph iff the flights can be scheduled using k gates. => If there is a schedule, the flights scheduled at the same gate have no conflict, and so we can colour the graph by using one colour for flights in each gate. <= If there is a graph colouring, then the vertices using each colour have no conflict, and so we can schedule the flights having the same colour in one gate.4/20/2020 Graph Theory 47
- 48. 257, 67 122,145 99 306 4 colors 4 gates assign gates: 257 67 99 145 306 122 Colouring the Vertices 4/20/2020 Graph Theory 48
- 49. Better Colouring 3 colors 3 gates 257 67 99 145 306 122 4/20/2020 Graph Theory 49
- 50. Homework#2: Final Exams subjects conflict if student takes both, so need different time slots. how short an exam period? Each vertex is a course, two courses have an edge if there is a conflict. The graph has a k-colouring if and only if the exams can be scheduled in k days. 4/20/2020 Graph Theory 50
- 51. Homework#3: Frequency Assignment Assign minimum number of frequencies to radio stations to avoid interference. 4/20/2020 Graph Theory 51
- 52. Homework#4: Register Allocation • Given a program, we want to execute it as quick as possible. • Calculations can be done most quickly if the values are stored in registers. • But registers are very expensive, and there are only a few in a computer. • Therefore we need to use the registers efficiently. 4/20/2020 Graph Theory 52
- 53. • Each node is a variable. • Two variables have a conflict if they cannot be put into the same register. a and b cannot use the same register, because they store different values. c and d cannot use the same register otherwise the value of c is overwritten. Each colour corresponds to a register. 4/20/2020 Graph Theory 53